Physics Laboratory Manual • Loyd L A B O R A T O R Y 21

Standing Waves on a String

O B J E C T I V E S • Demonstrate formation of standing waves on a string f r o m the interference of traveling waves

traveling i n opposite directions.

• Determine the tension T i n the string required to produce standing waves.

• Investigate the relationship between tension T and the wavelength X of the wave.

• Determine an experimental value for the frequency / of the wave and compare it to the k n o w n value of 120 H z .

E Q U I P M E N T L I S T • String vibrator (60 H z A C ) , string, clamps, pul ley, support rods, meter stick

• Mass holder, slotted masses

• Laboratory balance capable of measuring to 0.00001 kg (one for class)

T H E O R Y Waves are one means by w h i c h energy can be transported. Waves on a string are an example of transverse waves. These are waves i n w h i c h i n d i v i d u a l particles of the m e d i u m ( in this case the string) move perpendicular or transverse to energy m o v i n g along the string. In Figure 21-1 a string tied to a vibrator at one end passes over a pul ley, and the weight of masses on the other end provides tension T i n the string. The vibrator moves u p and d o w n at a frequency /, w h i c h causes a wave of that same frequency to propagate d o w n the string. The vibrator used i n this laboratory is dr iven by an electromagnet at a fre­quency of 60 H z , but because the electromagnet attracts the steel blade twice i n each cycle, the vibrator frequency is 120 H z .

The point at w h i c h the string passes over the pul ley is a f ixed point , and the wave is reflected f r o m that point . Thus the string is a m e d i u m i n w h i c h t w o waves of the same speed, frequency, and wavelength travel in the opposite direction. These t w o waves w i l l interfere w i t h each other to produce a standing wave w h e n the proper relationship exists between the string length L and the wavelength A of the wave. When a standing wave is produced, its characteristic features are the existence of nodes and antinodes at points along the string. A node N is a point for w h i c h there is no displacement of the string f r o m its equi l ibr ium posit ion. A n antinode A is a point on the string for w h i c h the ampli tude of vibrat ion is a m a x i m u m at all times. To f o r m a standing wave a node must occur at each end of the string, and an antinode must occur between each node. The distance between nodes is A/2 or one-half of a wavelength of the wave. I n terms of the string length L, a standing wave is possible w h e n

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218 Physics Laboratory Manual • Loyd

m

Figure 21-1 Experimental arrangement of vibrator, pulley, and masses.

L = n{X/l) where n = 1,2,3,4, (Eq. 1)

Figure 21-2 shows the first four standing waves that are possible. From the f igure i t is clear that n is the number of segments of half wavelengths i n each standing wave. Solving Equation 1 for the possible values of the wavelength X gives

/. = IL/n where n = 1,2,3,4, (Eq. 2)

Because the frequency is f ixed at 120 H z , each different standing wave w i l l have a different wave speed V. The wave speed is determined by the string tension T and the string mass per u n i t length p. The relationship between these quantities is given by

(Eq. 3)

The frequency, wavelength, and speed are related by V=//.. Using that and the speed V f r o m Equation 3 gives

The experimental arrangement differs sl ightly f r o m the ideal situation because the node at the vibrator end of the string is m o v i n g u p and d o w n rather than being fixed i n space. This effect means that the wavelength is somewhat di f f i cul t to define. For example, i n the case n = 2 the node w i l l not be exactly i n the center of the string, and each of the segments w i l l be sl ightly different i n length. The effect decreases

n = 1 n = 2

Laboratory 21 • Standing Waves on a String 219

w i t h each addit ion to the number of segments. One w a y to account for this effect is to determine the wavelength using only those segments that do not include the vibrator. Because i t is somewhat di f f i cul t to locate the nodes precisely, there is usually less error involved i f we assume that each wavelength is the total length of the string d i v i d e d by the number of segments. That is the assumption we w i l l make i n this laboratory.

E X P E R I M E N T A L P R O C E D U R E 1. The mass per u n i t length of the string should be determined by the class as a whole . Carefully measure

a 1.0000 m length of the type string to be used. Measure the mass of the string ms to the nearest 0.00001 kg . From these data f i n d p and record i t i n the Data Table.

2. Clamp the vibrator and the pul ley at opposite ends of the laboratory table approximately 1.5 m apart if the table w i l l a l low that. Tie one end of a piece of string to the vibrator and make a loop i n the other end that hangs over the pul ley as shown i n Figure 21-1. Place a 0.0500 k g mass holder on the loop.

3. Measure the length of the string f r o m the t ip of the vibrator to the point where the string touches the pul ley and record this as string length L i n the Data Table.

4. Plug i n the vibrator and place on the mass holder the mass required to produce a standing wave that corresponds to n = 3 i n Figure 21-2. Do not attempt the modes w i t h n — \d n = 2 because they require such h igh string tension that the string is l ikely to break. To determine the mass needed to produce a particular standing wave, i t w i l l help to p u l l d o w n sl ightly or l i f t u p sl ightly on the mass holder to determine if mass needs to be added or subtracted. Determine to the nearest 0.001 kg the mass needed to produce the largest possible ampli tude of v ibrat ion w i t h n = 3. Record the value of the mass M i n the Data Table for n = 3.

5. Remove mass and determine to the nearest 0.001 k g the mass needed to produce the largest ampli tude w i t h n = 4 and record that mass value i n the Data Table.

6. Continue the process of removing mass to produce standing waves for the cases of n = 5, 6, 7, 8, and 9. I n each case n refers to the number of segments into w h i c h the string length is d i v i d e d . For each case determine the mass needed (to the nearest 0.001 kg) to produce the largest ampli tude. Record al l values of mass i n the Data Table.

C A L C U L A T I O N S 1. Use the value of L i n Equation 2 to calculate the wavelength X for each of the standing waves n = 3

through n = 9 and record the results i n the Calculations Table.

2. From the measured values of M calculate the tension T = Mg w i t h g = 9.80 m / s 2 . Record the values of T i n the Calculations Table.

3. Calculate the values of VT for each of the values of T and record them i n the Calculations Table. Note that w h e n taking the square root, generally an extra significant f igure is a l lowed i n the value of the square root. For example, if a measured tension T were 7.83, then the recorded value of vT should be 2.798.

4. According to Equation 4, k should be proport ional to VT w i t h 1/fVP as the constant of pro­port ional i ty . Perform a linear least squares f i t w i t h k as the vertical axis and VT as the horizontal axis. Record the slope of the f i t i n the Calculations Table. Equate the slope to 1/fVP treating / as an u n k n o w n . Use the k n o w n value of p to solve the resulting equation for /. Record that value of the frequency as / e x p i n the Calculations Table. Also record the value of r the correlation coefficient i n the Calculations Table.

5. Calculate and record the percentage error of / e x p compared to the k n o w n value of the frequency / = 1 2 0 H z .

Name Section Date

Lab Partners

L A B O R A T O R Y 21 Standing Waves on a String

L A B O R A T O R Y R E P O R T

Data Table Calculations Table

n M ( k g )

3

4

5

6

7

8

9

p= k g / m L = m

T = Mg (N) A(m) Vf (VN)

Slope = r =

/ e x p Hz % Error =

S A M P L E C A L C U L A T I O N S 1. k = (2L)/n =

2. T = Mg —

3. Vf =

4. /exp = 1/(y/P Slope) =

5. % Error = \E - K\/K (100%) =

1 /T 6. (Question 3) f — - W— =

A U p